CH-9-10 Risk and Return: Lessons from Market History – Corporate Finance : Book Guide

This Document Contains Chapters 9 to 10 Chapter 9 Risk and Return: Lessons from Market History 1. A return is the change in value of an investment. A monetary return is the monetary change in the value of an investment, such as £2 or €2. A percentage return is the percentage change in the value of an investment, such as 12%. 2. This question is a trick question because a holding period return relates to how one calculates the change in value of the investment, whereas percentage return relates to how one expresses the change in value of an investment. A holding period return is the return an investor receives from holding an asset for a certain period. That is, the change in the value of the investment from the beginning to end of the period. The holding period return can be expressed in terms of monetary return or percentage return. 3. Return distributions succinctly presents the history of an investment’s returns over a period of time. Also, because returns are random variables, the most appropriate way to describe them is as a statistical distribution. Because distributions are in nominal terms, inflation would shift the distribution to the right. 4. Risk represents the possibility that the value of an investment can go up or down. Although in the long term, equities tend to outperform bonds, investors can be highly risk averse or have investment horizons that are quite short. 5. To calculate an arithmetic return, you simply sum the returns and divide by the number of returns. As such, arithmetic returns do not account for the effects of compounding. Geometric returns do account for the effects of compounding. As an investor, the more important return of an asset is the geometric return. 6. Although the index went up very quickly, the risk of investing in Venezuela was also exceptionally high. Risk averse investors would not consider such an investment because of the high likelihood of losses. 7. It’s easy to see after the fact that the investment was terrible, but it probably wasn’t so easy ahead of time. 8. This is a criticism that has been around for a number of years, especially in regards to exchange traded funds. It could be argued that unlike gambling, the stock market is a positive sum game; everybody can win. Also, speculators provide liquidity to markets and thus help to promote efficiency. However, the criticism is that ETFs are portfolios of stocks that can be traded. This means that investors become more concerned with the return and risk properties of the This Document Contains Chapters 9 to 10 portfolio rather than the business of the underlying assets. When trading is heavier in the ETF than in the equities that comprise the ETF, there is a danger that investors lose a connection with the actual purpose of the investment. In this case, it could be argued that ETFs are similar to gambling. 9. Before the fact, for most assets, the risk premium will be positive; investors demand compensation over and above the risk-free return to invest their money in the risky asset. After the fact, the observed risk premium can be negative if the asset’s nominal return is unexpectedly low, the risk-free return is unexpectedly high, or if some combination of these two events occurs. 10. Yes, the share prices are currently the same. Below is a diagram that depicts the equities’ price movements. Two years ago, each equity had the same price, P0. Over the first year, General Materials’ share price increased by 10 percent, or (1.1)  P0. Standard Fixtures’ share price declined by 10 percent, or (0.9)  P0. Over the second year, General Materials’ share price decreased by 10 percent, or (0.9)(1.1)  P0, while Standard Fixtures’ share price increased by 10 percent, or (1.1)(0.9)  P0. Today, each of the equities is worth 99 percent of its original value. 2 years ago 1 year ago Today General Materials P0 (1.1)P0 (1.1)(0.9)P0 = (0.99)P0 Standard Fixtures P0 (0.9)P0 (0.9)(1.1)P0 = (0.99)P0 11. Risk premiums are about the same whether or not we account for inflation. The reason is that risk premiums are the difference between two returns, so inflation essentially nets out. Returns, risk premiums, and volatility would all be lower than we estimated because aftertax returns are smaller than pretax returns. 12. a. The total euro return is the change in price plus the coupon payment, so: Total euro return = €1,074 – €1,200 + €80 Total euro return = -€46 b. The total percentage return of the bond is: R = [(€1,074 – €1,200) + €80] / €1,200 R = -.0383 or -3.83% Notice here that we could have simply used the total euro return of -€46 in the numerator of this equation. c. Using the Fisher equation, the real return was: (1 + R) = (1 + r)(1 + h) r = (1+(-.0383) / 1.030) – 1 r = -.0666 or -6.66% 13. The average return is the sum of the returns, divided by the number of returns. The average return for each equity was: RManGroup = rMianGroup, i=1 éåN êë ù úû N = ëé-.607 +.241+.336 -.623-.218ûù 5 = -.175 or -1.75% RITV = rIiTV , i=1 éåN êë ù úû N = ëé-.036 +.375+1.055 -.618 -.332ûù 5 = .0889 or 8.89% We calculate the variance of each equity as: s2 = (xi -x )2 i=1 éåN êë ù úû (N -1) sM anGroup 2 = 1 5-1 (-.607 -(-.087)) { 2 +(.241-(-.087))2 +(.336 -(-.087))2 +(-.623-(-.087))2 +(-.218 -(-.087))2} = .2343 sI2TV= 1 5-1 (-.036 -.0889) { 2 +(.375 -.0889)2 +(1.055 -.0889)2 +(-.618 -.0889)2 +(-.332 -.0889)2} = .426917 The standard deviation is the square root of the variance, so the standard deviation of each equity is: SManGroup = (.2343)1/2 SManGroup = .4841 or 48.41% SITV = (.426917)1/2 sY = .6534 or 65.34% 14. We first need to calculate the percentage returns for each country. The formula to use is: 1 1 t t t t P P r P − − − = Index Values: Year China Denmark France Germany India The Netherlands Norway Sweden Switzerland UK US Jan-05 100 100 100 100 100 100 100 100 100 100 100 Jan-06 91.67 137.04 124.21 131.37 147.36 124.72 161.23 129.27 134.97 117.36 107.95 Jan-07 211.24 158.97 139.39 158.93 228.27 134.3 221.07 158.86 161.31 131.46 121.49 Jan-08 415.44 145.35 115.21 148.45 346.86 116.79 201.38 122.74 127.22 122.47 115.49 Jan-09 148.49 95.36 64.09 83.55 152.59 53.84 123.58 90.34 76.1 86.136 69.84 Jan-10 256.12 124.73 91.51 128.72 285.9 85.59 178.09 126.46 111.23 110.06 93.2 Jan-11 225.23 164.96 100.6 160.87 330.17 96.9 211.29 154.06 112.91 126.16 109.4 Jan-12 184.22 136.74 79.31 137.39 281.77 81.82 189.93 130.3 99.99 116 106.31 Index Returns: Year China Denmark France Germany India The Netherlands Norway Sweden Switzerland UK US 2005 -0.083 0.370 0.242 0.314 0.474 0.247 0.612 0.293 0.350 0.174 0.080 2006 1.304 0.160 0.122 0.210 0.549 0.077 0.371 0.229 0.195 0.120 0.125 2007 0.967 -0.086 -0.173 -0.066 0.520 -0.130 -0.089 -0.227 -0.211 -0.068 -0.049 2008 -0.643 -0.344 -0.444 -0.437 -0.560 -0.539 -0.386 -0.264 -0.402 -0.297 -0.395 2009 0.725 0.308 0.428 0.541 0.874 0.590 0.441 0.400 0.462 0.278 0.334 2010 -0.121 0.323 0.099 0.250 0.155 0.132 0.186 0.218 0.015 0.146 0.174 2011 -0.182 -0.171 -0.212 -0.146 -0.147 -0.156 -0.101 -0.154 -0.114 -0.081 -0.028 a. The average return for each country’s stock market over this period was: China Denmark France Germany India The Netherlands Norway Sweden Switzerland UK US Average Return 0.281 0.080 0.009 0.095 0.266 0.032 0.148 0.071 0.042 0.039 0.034 b. Using the equation for variance, we find the variance for each country over this period 2 ( )2 ( ) 1 1 N x i i s x x N =   =  −  −    China Denmark France Germany India The Netherlands Norway Sweden Switzerland UK US Average Return 0.281 0.080 0.009 0.095 0.266 0.032 0.148 0.071 0.042 0.039 0.034 Variance 0.513 0.079 0.090 0.108 0.237 0.126 0.126 0.076 0.097 0.039 0.053 Standard Deviation 0.716 0.280 0.299 0.329 0.487 0.355 0.355 0.276 0.311 0.197 0.229 15. The annual returns for BMW must first be calculated using the return formula. 1 1 t t t t t P Div P r P − − + − = Year Open Price Dividend Return 2003 €26.87 0.52 32.90% 2004 €35.19 0.58 -6.93% 2005 €32.17 0.62 17.66% 2006 €37.23 0.64 27.53% 2007 €46.84 0.7 -19.94% 2008 €36.80 0 -49.43% 2009 €18.61 0 66.36% 2010 €30.96 0.3 82.11% 2011 €56.08 0.3 17.14% 2012 €65.39 0 The average return is 0.186 or 18.6%. b. The variance of BMW’s return is calculated using the variance formula. 2 ( )2 ( ) 1 1 N x i i s x x N =   =  −  −    This is 0.167085. The standard deviation is the square root of the variance and is 0.4202 or 40.02%. 16. To calculate the average real return, we can use the average return of the asset and the average inflation rate in the Fisher equation. Doing so, we find: (1 + R) = (1 + r)(1 + h) r = (1.186/1.042) – 1 r = .138 or 13.8% b. The average nominal risk premium is simply the average nominal return of the asset, minus the average nominal risk-free rate, so, the average nominal risk premium for this asset would be: RP = R – R f RP = .186 – .051 RP = 13.5% c. Using the Fisher equation, we can calculate the average real risk-free rate as: (1 ) 1.051 1 1 0.86% (1 ) 1.042 f f R r h + = − = − = + The average real risk premium is simply the average real return of the asset, minus the average real risk-free rate, so, the average real risk premium for this asset would be: rp = r -rf =13.8% -0.86% =12.94% 17. We must first find the returns on the company. Year Price Dividend Return 2008 £1.12 0 - 2009 £1.34 0 0.196 2010 £1.68 0 0.254 2011 £1.8825 0 0.121 2012 £2.18 0 0.158 2013 £2.07 0 -0.050 Apply the five-year holding-period return formula to calculate the total return of the equity over the five-year period, we find: 5-year holding-period return = [(1 + R1)(1 + R2)(1 +R3)(1 +R4)(1 +R5)] – 1 5-year holding-period return = [(1 – .196)(1 + .254)(1 + .121)(1 + .158)(1 + -.050)] – 1 5-year holding-period return = 0.8482 or 84.82%. 18. Looking at the long-term European return history in Table 9.3, we see that the mean arithmetic return was 5.2 per cent, with a standard deviation of 16.6 per cent. The range of returns you would expect to see 68 percent of the time is the mean plus or minus 1 standard deviation, or: R  ± 1 = 5.2% ± 16.6% = –11.4% to 21.8% The range of returns you would expect to see 95 percent of the time is the mean plus or minus 2 standard deviations, or: R  ± 2 = 5.2% ± 2(16.6%) = –28% to 38.4% 19. From Table 9.3, The arithmetic return is (1.6 + 1.2 = ) 2.8 per cent. The geometric return is (3.9 + 1.2 = ) 5.1 per cent. Table 9.3 is based on 109 years of data, so N = 109. T is the average return forecast. To find the best forecast for other periods, we apply Blume’s formula as follows: R(1) = 1 - 1 109 - 1 × 5.1% + 109 - 1 109 - 1 × 2.8% = 2.8% R(5) = 5 - 1 109 - 1 × 5.1% + 109 - 5 109 - 1 × 2.8% = 2.89% R(20) = 20 - 1 109 - 1 × 5.1% + 109 - 20 109 - 1 × 2.8% = 3.20% R(30) = 30 - 1 109 - 1 × 5.1% + 109 - 30 109 - 1 × 2.8% = 3.42% 20. The arithmetic average return is the sum of the known returns divided by the number of returns, so: Arithmetic average return = (.19 + .17 + .21 –.08 + .09 –.14) / 6 Arithmetic average return = .073 or 7.3% Using the equation for the geometric return, we find: Geometric average return = [(1 + R1) × (1 + R2) × … × (1 + RT)]1/T – 1 Geometric average return = [(1 + .19)(1 + .17)(1 + .21)(1 – .08)(1 + .09)(1 – .14)](1/6) – 1 Geometric average return = .0642 or 6.42% Remember, the geometric average return will always be less than the arithmetic average return if the returns have any variation. 21. To calculate the arithmetic and geometric average returns, we must first calculate the return for each year. The return for each year is: R1 = (£49.07 – 43.12 + 0.55) / £43.12 = .1507 or 15.07% R2 = (£51.19 – 49.07 + 0.60) / £49.07 = .0554 or 5.54% R3 = (£47.24 – 51.19 + 0.63) / £51.19 = –.0649 or –6.49% R4 = (£56.09 – 47.24 + 0.72)/ £47.24 = .2026 or 20.26% R5 = (£67.21 – 56.09 + 0.81) / £56.09 = .2127 or 21.27% The arithmetic average return was: RA = (0.1507 + 0.0554 – 0.0649 + 0.2026 + 0.2127)/5 RA = 0.1113 or 11.13% And the geometric average return was: RG = [(1 + .1507)(1 + .0554)(1 – .0649)(1 + .2026)(1 + .2127)]1/5 – 1 RG = 0.1062 or 10.62% 22. To find the return on the coupon bond, we first need to find the price of the bond today. Since one year has elapsed, the bond now has six years to maturity, so the price today is: P1 = NOK80(PVIFA7%,6) + NOK1,000/1.076 P1 = NOK1,047.67 You received the coupon payments on the bond, so the nominal return was: R = (NOK1,047.67 – 1,028.50 + 80) / NOK1,028.50 R = .0964 or 9.64% And using the Fisher equation to find the real return, we get: r = (1.0964 / 1.048) – 1 r = .0462 or 4.62% 23. The mean return was 5.8 per cent, with a standard deviation of 9.3 per cent. In the normal probability distribution, approximately 2/3 of the observations are within one standard deviation of the mean. This means that 1/3 of the observations are outside one standard deviation away from the mean. Or: Pr(R15.1)  1/3 But we are only interested in one tail here, that is, returns less than –3.5 percent, so: Pr(R< –3.5)  1/6 You can use the z-statistic and the cumulative normal distribution table to find the answer as well. Doing so, we find: z = (X – µ)/ z = (–3.5% – 5.8)/9.3% = –1.00 Looking at the z-table, this gives a probability of 15.87%, or: Pr(R< –3.5)  .1587 or 15.87% The range of returns you would expect to see 95 percent of the time is the mean plus or minus 2 standard deviations, or: 95% level: R=  ± 2 = 5.8% ± 2(9.3%) = –12.80% to 24.40% The range of returns you would expect to see 99 percent of the time is the mean plus or minus 3 standard deviations, or: 99% level: R=  ± 3 = 5.8% ± 3(9.3%) = –22.10% to 33.70% 24. The mean return for French shares was 0.9 per cent, with a standard deviation of 29.9 per cent. Doubling your money is a 100% return, so if the return distribution is normal, we can use the z-statistic. So: z = (X – µ)/ z = (100% – 0.9%)/29.9% = 3.3144 standard deviations above the mean This corresponds to a probability of  0.08%, or less than once every 1,200 years. Tripling your money would be: z = (200% – 0.9%)/29.9% = 6.6589 standard deviations above the mean. This corresponds to a probability of (much) less than 0.08%. The actual answer is once every 2.13 x 1010 years. 25. It is impossible to lose more than 100 per cent of your investment. Therefore, return distributions are truncated on the lower tail at –100 per cent. 26. Using the z-statistic, we find: z = (X – µ)/ z = (0% – 7.1%)/27.6% = –0.25725 Pr(R=0)  19.30% 27. The return series is as follows: R = (4 – 0.9)/10 = .31 = 31% The formula for variance is given below: 2 ( )2 ( ) 1 1 N x i i s x x N =   =  −  −    Using this formula, the variance is equal to 1.761 and the standard deviation is equal to 1.327 or 132.7% 28. To calculate real returns, we use the formula: (1 + real rate) = (1 + nominal rate)/(1 + inflation rate) The annual real returns for Man Group and ITV are given below: Year Man Group Nominal ITV Nominal inflation Man Group Real ITV Real 2011 -60.70% -3.60% 4.80% -62.50% -8.02% 2010 24.10% 37.50% 4.80% 18.42% 31.20% 2009 33.60% 105.50% 2.40% 30.47% 100.68% 2008 -62.30% -61.80% 0.90% -62.64% -62.14% 2007 -21.80% -33.20% 4.00% -24.81% -35.77% The average real return for Man Group is 20.21% and the average real return for ITV is 5.19%. 29. This exercise is for students to carry out themselves. It is designed to get them used to collecting their own data and undertaking their own analysis. The calculations are very simple and the test is the ability to do independent research. 30. First, the returns of Banco Santander must be calculated: Date Adj Close Returns 01/03/2012 514.2 -0.73% 01/02/2012 518 5.00% 03/01/2012 493.35 0.12% 01/12/2011 492.75 4.18% 01/11/2011 473 -11.26% 03/10/2011 533 -1.11% 01/09/2011 539 -4.43% 01/08/2011 564 -11.88% 01/07/2011 640 -10.49% 01/06/2011 715 -0.69% 03/05/2011 720 -5.64% 01/04/2011 763 6.86% 01/03/2011 714 -5.80% 01/02/2011 758 -0.85% 04/01/2011 764.5 11.52% 01/12/2010 685.5 12.29% 01/11/2010 610.5 -23.33% 18/10/2010 796.24 The mean return is -2.13 per cent and the standard deviation of return 9.03 per cent. To find the monthly VaR with a 1 per cent loss of probability : VaR = 2.13 – 2.33(9.03) = -23.1827% With a €1 million investment, the amount at risk is €231,827. Chapter 10 Risk and Return: The Capital Asset Pricing Model 1. The three characteristics are Expected Return, Variance, and Covariance. Expected return tells you how much you expect to get on an investment, variance tells you the precision of your estimate and covariance tells you how your investment interacts with other investments in your portfolio. 2. Correlation is the standardised covariance between two asset returns. The formula for covariance is: Correlation is related to covariance and variance through the numerator and denominator in its formula. 3. The best way to explain diversification is by using the phrase, ‘don’t put all your eggs in one basket’ or something similar. By spreading your eggs across many baskets, you lower the risk that if you drop one basket, all the eggs will be ruined. 4. A minimum variance portfolio is the portfolio that has the minimum possible risk. It is possible for two portfolios to have the same risk but different expected returns due to the mathematical properties of investment portfolios. The diagram below shows how this can happen. Cov( , ) Corr( , ) SD( ) SD( ) A B AB A B A B R R R R  = = R  R The vertical line dropping from the point 2, crosses the minimum variance frontier next to point 1. Both portfolios have the same risk but different expected returns. 5. It is an impossible scenario unless all the securities in the portfolio had zero risk, which would mean that they were all risk free. 6. Covariance becomes more important because as the number of securities in a portfolio increases, the number of covariance terms grows by significantly more in numbers than variance terms. This is illustrated below: Number of Number of Number of Total Variance Terms Covariance Terms Securities in Number of (number of terms (number of terms Portfolio Terms on diagonal) off diagonal) 1 1 1 0 2 4 2 2 3 9 3 6 10 100 10 90 100 10,000 100 9,900 . . . . . . . . . . . . N N2 N N2 – N 7. The optimal portfolio is the feasible portfolio that, in conjunction with the risk free asset, can deliver the highest expected return for the lowest level of risk. It is the portfolio that has the highest reward to risk ratio of every possible capital allocation line. The conditions are: riskless borrowing and lending, everyone having the same views on expected returns and risk of all assets, no transaction costs and the existence of a risk free asset. Although these are not likely to be applicable in the real world, the theory provides many insights into how assets are priced. 8. The wide fluctuations in the price of oil shares do not indicate that these are a poor investment. If an oil share is purchased as part of a well-diversified portfolio, only its contribution to the risk of the entire portfolio matters. This contribution is measured by systematic risk or beta. Since price fluctuations in oil shares reflect diversifiable plus non- diversifiable risk, observing the standard deviation of price movements is not an adequate measure of the appropriateness of adding oil shares to a portfolio. 9. Yes. It is possible, in theory, to construct a zero beta portfolio of risky assets whose return would be equal to the risk-free rate. It is also possible to have a negative beta; the return would be less than the risk-free rate. A negative beta asset would carry a negative risk premium because of its value as a diversification instrument. 10. Roll argues that the CAPM is untestable because it is impossible to find a portfolio that truly represents the market portfolio. Since this is unobservable, any test of the CAPM is a test of the proxy portfolio instead of the theory. 11. The consumption CAPM and the Human Capital CAPM are both extensions of the basic CAPM model. They are not very popular with practitioners because it is very difficult to get reliable data on consumption growth or human capital. 12. The investment manager is confusing total risk with systematic risk. Although Modern Times Group AB has more volatile returns, much of this may be diversifiable. Beta, which is systematic risk, measures that risk which cannot be diversified. 13. This is a market timing strategy: Invest in aggressive (high beta) stocks when the market is rising and defensive (low beta) stocks when the market is falling. From a risk management perspective, investing in low beta stocks hedges against downside risk at the cost of losing out on upside movements. By moving in and out of low and high beta securities depending on market conditions, one can maximize return potential. 14. The validity of the broker’s advice depends on whether you have a diversified portfolio or whether you only have one or two equity investments. If your investment portfolio is diversified, much of the volatility may be diversified away. However, if you only hold one or two securities, volatility in one stock may result in acceptably high risk. 15. a. We have a special case where the portfolio is equally weighted, so we can sum the returns of each asset and divide by the number of assets. The expected return of the portfolio is: E(Rp) = (.09 + .02)/2 = .055 or 5.50% b. We need to find the portfolio weights that result in a portfolio with a  of 0.6. We know the  of the risk-free asset is zero. We also know the weight of the risk-free asset is one minus the weight of the equity since the portfolio weights must sum to one, or 100 percent. So: p = 0.6 = wS(0.9) + (1 – wS)(0) 0.6 = 0.9wS + 0 – 0wS wS = 0.6/0.9 wS = .666667 And, the weight of the risk-free asset is: wRf = 1 – .66667 = .33333 c. We need to find the portfolio weights that result in a portfolio with an expected return of 6 percent. We also know the weight of the risk-free asset is one minus the weight of the stock since the portfolio weights must sum to one, or 100 percent. So: E(Rp) = .06 = .09wS + .02(1 – wS) .06 = .09wS + .02 – .02wS wS = .5714 So, the risk of the portfolio will be: p = .5714(.9) + (1 – .5714)(0) = 0.514 d. Solving for the  of the portfolio as we did in part a, we find: p = 1.5 = wS(0.9) + (1 – wS)(0) wS = 1.5/.9 = 1.667 wRf = 1 – 1.667 = –0.667 The portfolio is invested 167% in the equity and –67% in the risk-free asset. This represents borrowing at the risk-free rate to buy more of the equity. 16. We need to find the portfolio weights that result in a portfolio with an expected return of 6 percent. We also know the weight of the risk-free asset is one minus the weight of the stock since the portfolio weights must sum to one, or 100 percent. So: E(Rp) = .05 = .09wS + .03(1 – wS) .05 = .09wS + .03 – .03wS wS = .3333 So, the risk of the portfolio will be:  = ws (M) = .3333 (.23) = 7.667% 17. First, we need to find the  of the portfolio. The  of the risk-free asset is zero, and the weight of the risk-free asset is one minus the weight of the equity, the  of the portfolio is: ßp = wW(1.2) + (1 – wW)(0) = 1.2wW So, to find the  of the portfolio for any weight of the equity, we simply multiply the weight of the equity times its . Even though we are solving for the  and expected return of a portfolio of one equity and the risk-free asset for different portfolio weights, we are really solving for the SML. Any combination of this equity, and the risk-free asset will fall on the SML. For that matter, a portfolio of any equity and the risk-free asset, or any portfolio of equities, will fall on the SML. We know the slope of the SML line is the market risk premium, so using the CAPM and the information concerning this equity, the market risk premium is: E(RW) = .12 = .03 + MRP(1.20) MRP = .09/1.2 = .075 or 7.5% So, now we know the CAPM equation for any equity is: E(Rp) = .03 + .075p The slope of the SML is equal to the market risk premium, which is 0.75. Using these equations to fill in the table, we get the following results: ww E(Rp) Bp 0% 0% 0 25% 25% 0.3 50% 50% 0.6 75% 75% 0.9 100% 100% 1.2 125% 125% 1.5 150% 150% 1.8 18. There are two ways to correctly answer this question. We will work through both. First, we can use the CAPM. Substituting in the value we are given for each stock, we find: E(RY) = .055 + .075(1.50) = .1675 or 16.75% It is given in the problem that the expected return of Y is 17 percent, but according to the CAPM, the return of the equity based on its level of risk should be 16.75 percent. This means the equity return is too high, given its level of risk. Equity Y plots above the SML and is undervalued. In other words, its price must increase to reduce the expected return to 16.75 percent. For Equity Z, we find: E(RZ) = .055 + .075(0.80) = .1150 or 11.50% The return given for Z is 10.5 percent, but according to the CAPM the expected return of the equity should be 11.50 percent based on its level of risk. Equity Z plots below the SML and is overvalued. In other words, its price must decrease to increase the expected return to 11.50 percent. We can also answer this question using the reward-to-risk ratio. All assets must have the same reward-to-risk ratio, that is, every asset must have the same ratio of the asset risk premium to its beta. This follows from the linearity of the SML in Figure 10.11. The reward-to-risk ratio is the risk premium of the asset divided by its . This is also known as the Treynor ratio or Treynor index. We are given the market risk premium, and we know the  of the market is one, so the reward-to-risk ratio for the market is 0.075, or 7.5 percent. Calculating the reward- to-risk ratio for Y, we find: Reward-to-risk ratio Y = (.17 – .055) / 1.50 = .0767 The reward-to-risk ratio for Y is too high, which means the equity plots above the SML, and the equity is undervalued. Its price must increase until its reward-to-risk ratio is equal to the market reward-to-risk ratio. For equity Z, we find: Reward-to-risk ratio Z = (.105 – .055) / .80 = .0625 The reward-to-risk ratio for Z is too low, which means the equity plots below the SML, and the equity is overvalued. Its price must decrease until its reward-to-risk ratio is equal to the market reward-to-risk ratio. We now need to set the reward-to-risk ratios of the two assets equal to each other which is: (.17 – Rf)/1.50 = (.105 – Rf)/0.80 We can cross multiply to get: 0.80(.17 – Rf) = 1.50(.105 – Rf) Solving for the risk-free rate, we find: 0.136 – 0.80Rf = 0.1575 – 1.50Rf Rf = .0307 or 3.07% 19. The expected return of a portfolio is the sum of the weight of each asset times the expected return of each asset. So, the expected return of the portfolio is: E(Rp) = .40(.15) + .60(.16) = .156 or 15.6% The variance is: The standard deviation is the square root of the variance: Standard Deviation = 0.3264 or 32.64% 20. We know the total portfolio value and the investment of two equities in the portfolio, so we can find the weight of these two equities. The weights of equity A and equity B are: wA = €200,000 / €1,000,000 = .20 wB = €250,000/€1,000,000 = .25 Since the portfolio is as risky as the market, the  of the portfolio must be equal to one. We also know the  of the risk-free asset is zero. We can use the equation for the  of a portfolio to find the weight of the third equity. Doing so, we find: p = 1.0 = wA(.8) + wB(1.3) + wC(1.5) + wRf(0) 2 2 2 2 2 2 2 2 2 2 2 2 Var(portfolio) 2 2 .4 .3 .6 .4 2(.4)(.6)(.3)(.4)(.6) .10656 A A A B AB B B A A A B A B AB B B X X X X X X X  X =  +  +  =  +   +  = + + = Solving for the weight of equity C, we find: wC = .343333 So, the euro investment in equity C must be: Invest in equity C = .343333(€1,000,000) = €343,333 We also know the total portfolio weight must be one, so the weight of the risk-free asset must be one minus the asset weight we know, or: 1 = wA + wB + wC + wRf 1 = .20 + .25 + .34333 + wRf wRf = .206667 So, the euro investment in the risk-free asset must be: Invest in risk-free asset = .206667(€1,000,000) = €206,667 21. We are given the expected return and  of a portfolio and the expected return and  of the risky assets in the portfolio. We know the  of the risk-free asset is zero. We also know the sum of the weights of each asset must be equal to one. So, the weight of the risk-free asset is one minus the weight of X and the weight of Y. Using this relationship, we can express the expected return of the portfolio as: E(Rp) = .085 = wX(.12) + wY(.09) + (1 – wX – wY)(.03) And the  of the portfolio is: p = .7 = wX(1.5) + wY(1.2) + (1 – wX – wY)(0) We have two equations and two unknowns. Solving these equations, we find that: wX = 1.33333 wY = -1.08333 wRf = 0.75 The amount to invest in X is: Investment in X = 1.33333(£24,000) = £31,999.92 A portfolio weight greater than 1 means that you have borrowed at the risk free rate or short sold the other security. If you are not familiar with short selling, it means you borrow a equity today and sell it. You must then purchase the equity at a later date to repay the borrowed equity. If you short sell an equity, you make a profit if the equity decreases in value. 22. The expected return of an asset is the sum of the probability of each return occurring times the rate of return. So, the expected return of each equity is: E(RA) = .33(.063) + .33(.105) + .33(.156) = .1080 or 10.80% E(RB) = .33(–.037) + .33(.064) + .33(.253) = .0933 or 9.33% To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the variance and standard deviation of equity A are: =.33(.063 – .1080)2 + .33(.105 – .1080)2 + .33(.156 – .1080)2 = .00145 = (.00145)1/2 = .0380 or 3.80% And the standard deviation of equity B is: =.33(–.037 – .0933)2 + .33(.064 – .0933)2 + .33(.253 – .0933)2 = .01445 = (.01445)1/2 = .1202 or 12.02% To find the covariance, we multiply probability of each possible state with the product of each assets’ deviation from the mean in that state. The sum of these products is the covariance. So, the covariance is: Cov(A,B) = .33(.063 – .1080)(–.037 – .0933) + .33(.105 – .1080)(.064 – .0933) + .33(.156 – .1080)(.253 – .0933) Cov(A,B) = .004539 And the correlation is: A,B = Cov(A,B) / A B A,B = .004539 / (.0380)(.1202) A,B = .9937 23. The expected return of an asset is the sum of the probability of each return occurring times the rate of return. So, the expected return of each equity is: E(RJ) = .25(–.020) + .60(.092) + .15(.154) = .0733 or 7.33% E(RK) = .25(.050) + .60(.062) + .15(.074) = .0608 or 6.08% To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the variance and standard deviation of equity J are:  =.25(–.020 – .0733)2 + .60(.092 – .0733)2 + .15(.154 – .0733)2 = .00336 2 A A 2 B B 2 J J = (.00336)1/2 = .0580 or 5.80% And the standard deviation of equity K is:  =.25(.050 – .0608)2 + .60(.062 – .0608)2 + .15(.074 – .0608)2 = .00006 K = (.00006)1/2 = .0075 or 0.75% To find the covariance, we multiply probability of each possible state with the product of each assets’ deviation from the mean in that state. The sum of these products is the covariance. So, the covariance is: Cov(J,K) = .25(–.020 – .0733)(.050 – .0608) + .60(.092 – .0733)(.062 – .0608) + .15(.154 – .0733)(.074 – .0608) Cov(J,K) = .000425 And the correlation is: J,K = Cov(J,K) / J K J,K = .000425 / (.0580)(.0075) J,K =0.98 24. a. The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset, so: E(RP) = wAE(RA) + wBE(RB) E(RP) = .40(.15) + .60(.25) E(RP) = .2100 or 21.00% The variance of a portfolio of two assets can be expressed as:  = w  + w  + 2wAwBABA,B  = .402(.402) + .602(.652) + 2(.40)(.60)(.40)(.65)(.50)  = .24010 So, the standard deviation is:  = (.24010)1/2 = .4900 or 49.00% b. The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset, so: E(RP) = wAE(RA) + wBE(RB) E(RP) = .40(.15) + .60(.25) E(RP) = .2100 or 21.00% 2 K 2 P 2 A 2 A 2 B 2 B 2 P 2 P The variance of a portfolio of two assets can be expressed as:  = w  + w  + 2wAwBABA,B  = .402(.402) + .602(.652) + 2(.40)(.60)(.40)(.65)(–.50)  = .11530 So, the standard deviation is:  = (.11530)1/2 = .3396 or 33.96% c. As A and B become less correlated, or more negatively correlated, the standard deviation of the portfolio decreases. 25. a. (i) We can use the equation to calculate beta, we find: A = (A,M)(A) / M 0.9 = (A,M)(0.38) / 0.20 A,M = 0.47 (ii) Using the equation to calculate beta, we find: B = (B,M)(B) / M 1.1 = (.40)(B) / 0.20 B = 0.55 (iii) Using the equation to calculate beta, we find: C = (C,M)(C) / M C = (.35)(.65) / 0.20 C = 1.14 (iv) The market has a correlation of 1 with itself. (v) The beta of the market is 1. (vi) The risk-free asset has zero standard deviation. (vii) The risk-free asset has zero correlation with the market portfolio. (viii) The beta of the risk-free asset is 0. b. Using the CAPM to find the expected return of the equity, we find: 2 P 2 A 2 A 2 B 2 B 2 P 2 P Firm A: E(RA) = Rf + A[E(RM) – Rf] E(RA) = 0.05 + 0.9(0.15 – 0.05) E(RA) = .1400 or 14.00% According to the CAPM, the expected return on Firm A’s equity should be 14 percent. However, the expected return on Firm A’s equity given in the table is only 13 percent. Therefore, Firm A’s equity is overpriced, and you should sell it. Firm B: E(RB) = Rf + B[E(RM) – Rf] E(RB) = 0.05 + 1.1(0.15 – 0.05) E(RB) = .1600 or 16.00% According to the CAPM, the expected return on Firm B’s equity should be 16 percent. The expected return on Firm B’s equity given in the table is also 16 percent. Therefore, Firm B’s equity is correctly priced. Firm C: E(RC) = Rf + C[E(RM) – Rf] E(RC) = 0.05 + 1.14(0.15 – 0.05) E(RC) = .1638 or 16.38% According to the CAPM, the expected return on Firm C’s equity should be 16.38 percent. However, the expected return on Firm C’s equity given in the table is 25 percent. Therefore, Firm C’s equity is underpriced, and you should buy it. 26. Because a well-diversified portfolio has no unsystematic risk, this portfolio should lie on the Capital Market Line (CML). The slope of the CML equals: SlopeCML = [E(RM) – Rf] / M SlopeCML = (0.12 – 0.05) / 0.10 SlopeCML = 0.70 a. The expected return on the portfolio equals: E(RP) = Rf + SlopeCML(P) E(RP) = .05 + .70(.07) E(RP) = .0990 or 9.90% b. The expected return on the portfolio equals: E(RP) = Rf + SlopeCML(P) .20 = .05 + .70(P) P = .2143 or 21.43% 27. a. The CAC 40 is the market index and must have a beta of 1. This allows us to calculate the slope of the security market line. S = (14% – 3%)/(1 – 0) = 11% This then allows us to calculate the betas of Publicis and Renault. Publicis: 11% = (17% – 3%)/βPublicis βPublicis = 1.27 Renault: 11% = (10% – 3%)/ βRenault βRenault = 0.6364 The risk of a portfolio with an expected return equal to the market return must also equal 1, the risk of the market. We will now show this in detail. First, find the relative weights of Publicis and Renault in a portfolio with a return of 14% 14% = w(17%) + (1-w)10% w = 0.5714 The beta of the portfolio is equal to: βP = 0.5714(1.27) + 0.4286(0.6364) = 1 b. Find the weights of the CAC 40 and risk free rate: 10% = w(14%) + (1-w)(3%) w = 0.6364 The beta of the portfolio is equal to 0.6364 and the standard deviation of the portfolio is (0.6364*17%=) 10.82%. As expected, the beta of Renault is the same as the beta of the portfolio. However, the standard deviation of the market portfolio is about 85% of the risk of Renault. This reflects the lack of systematic risk in the CAC 40. 28. a. The formula for the variance of a 3 asset portfolio is: The standard deviation is thus 16.46% b. Since the correlation between Afgri and Harmony Gold is not -1, it is impossible to have a combination of weights that will bring the portfolio standard deviation to zero. X A2s2A + XB2s2B + XC2sC2 +2X A XBsAsBrAB +2X A XCsAsC rAC +2XB XCsBsC rBC =.32(292) +.42(212) +.32(232) +2(.3)(.4)(29)(21)(-.2) +2(.3)(.3)(29)(23)(.5) +2(.4)(.3)(21)(23)(.4) = 271.026 29. The amount of systematic risk is measured by the  of an asset. Since we know the market risk premium and the risk-free rate, if we know the expected return of the asset we can use the CAPM to solve for the  of the asset. The expected return of Security I is: E(RI) = .15(.09) + .70(.42) + .15(.26) = .3465 or 34.65% Using the CAPM to find the  of Security I, we find: .3465 = .04 + 0.1I I =3.07 The total risk of the asset is measured by its standard deviation, so we need to calculate the standard deviation of Security I. Beginning with the calculation of the security’s variance, we find: I2 = .15(.09 – .3465)2 + .70(.42 – .3465)2 + .15(.26 – .3465)2 I2 = .01477 I = (.01477)1/2 = .1215 or 12.15% Using the same procedure for Security II, we find the expected return to be: E(RII) = .15(–.30) + .70(.12) + .15(.44) = .1050 Using the CAPM to find the  of Security II, we find: .1050 = .04 + 0.1II II = 0.65 And the standard deviation of Security II is: II2 = .15(–.30 – .105)2 + .70(.12 – .105)2 + .15(.44 – .105)2 II2 = .04160 II = (.04160)1/2 = .2039 or 20.39% Although Security II has more total risk than I, it has much less systematic risk, since its beta is much smaller than I’s. Thus, it has more systematic risk, and II has more unsystematic and more total risk. Since unsystematic risk can be diversified away, I is actually the “riskier” security despite the lack of volatility in its returns. Security I will have a higher risk premium and a greater expected return. 30. The Capital Market Line is the set of possible investments linking the risk free asset and the market portfolio in Expected Return – Standard Deviation Space. The Security Market Line is the line linking the risk free asset with the market portfolio in Expected Return – Beta Space. See Figures 10.9 and 10.11 for detailed graphs of the CML and SML. 31. Here we have the expected return and beta for two assets. We can express the returns of the two assets using CAPM. Now we have two equations and two unknowns. Going back to Algebra, we can solve the two equations. We will solve the equation for Renewable Energy Corp. to find the risk-free rate, and solve the equation for STATOIL to find the expected return of the market. We next substitute the expected return of the market into the equation for Renewable Energy, and then solve for the risk-free rate. Now that we have the risk-free rate, we can substitute this into either original CAPM expression and solve for expected return of the market. Doing so, we get: E(RRenewable) = .23 = Rf + 1.3(RM – Rf); E(RSTATOIL) = .13 = Rf + .6(RM – Rf) .23 = Rf + 1.3RM – 1.3Rf = 1.3RM – .3Rf; .13 = Rf + .6(RM – Rf) = Rf + .6RM – .6Rf Rf = (1.3RM – .23)/.3 RM = (.13 – .4Rf)/.6 RM = .217 – .667Rf Rf = [1.3(.217 – .667Rf) – .23]/.3 1.167Rf = .0517 Rf = .0443 or 4.43% .23 = .0443 + 1.3(RM – .0443) .13 = .0443 + .6(RM – .0443) RM = .1871 or 18.71% RM = .1871 or 18.71% 32. Students should refer to section 10.7 for a full discussion of the issues in this question. 33. a. The expected return of an asset is the sum of the probability of each return occurring times rate of return. To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the expected return and standard deviation of each security are: Asset 1: E(R1) = .10(.25) + .40(.20) + .40(.15) + .10(.10) = .1750 or 17.50%  =.10(.25 – .1750)2 + .40(.20 – .1750)2 + .40(.15 – .1750)2 + .10(.10 – .1750)2 = .00163 1 = (.00163)1/2 = .0403 or 4.03% Asset 2: E(R2) = .10(.25) + .40(.15) + .40(.20) + .10(.10) = .1750 or 17.50%  =.10(.25 – .1750)2 + .40(.15 – .1750)2 + .40(.20 – .1750)2 + .10(.10 – .1750)2 = .00163 2 = (.00163)1/2 = .0403 or 4.03% Asset 3: E(R3) = .10(.10) + .40(.15) + .40(.20) + .10(.25) = .1750 or 17.50%  =.10(.10 – .1750)2 + .40(.15 – .1750)2 + .40(.20 – .1750)2 + .10(.25 – .1750)2 = .00163 3 = (.00163)1/2 = .0403 or 4.03% b. To find the covariance, we multiply probability of each possible state with the product of each assets’ deviation from the mean in that state. The sum of these products is the 2 1 2 2 2 3 covariance. The correlation is the covariance divided by the product of the two standard deviations. So, the covariance and correlation between each possible set of assets are: Asset 1 and Asset 2: Cov(1,2) = .10(.25 – .1750)(.25 – .1750) + .40(.20 – .1750)(.15 – .1750) + .40(.15 – .1750)(.20 – .1750) + .10(.10 – .1750)(.10 – .1750) Cov(1,2) = .000625 1,2 = Cov(1,2) / 1 2 1,2 = .000625 / (.0403)(.0403) 1,2 = .3846 Asset 1 and Asset 3: Cov(1,3) = .10(.25 – .1750)(.10 – .1750) + .40(.20 – .1750)(.15 – .1750) + .40(.15 – .1750)(.20 – .1750) + .10(.10 – .1750)(.25 – .1750) Cov(1,3) = –.001625 1,3 = Cov(1,3) / 1 3 1,3 = –.001625 / (.0403)(.0403) 1,3 = –1 Asset 2 and Asset 3: Cov(2,3) = .10(.25 – .1750)(.10 – .1750) + .40(.15 – .1750)(.15 – .1750) + .40(.20 – .1750)(.20 – .1750) + .10(.10 – .1750)(.25 – .1750) Cov(2,3) = –.000625 2,3 = Cov(2,3) / 2 3 2,3 = –.000625 / (.0403)(.0403) 2,3 = –.3846 c. The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset, so, for a portfolio of Asset 1 and Asset 2: E(RP) = w1E(R1) + w2E(R2) E(RP) = .50(.1750) + .50(.1750) E(RP) = .1750 or 17.50% The variance of a portfolio of two assets can be expressed as:  = w  + w  + 2w1w2121,2  = .502(.04032) + .502(.04032) + 2(.50)(.50)(.0403)(.0403)(.3846)  = .001125 And the standard deviation of the portfolio is: P = (.001125)1/2 P = .0335 or 3.35% 2 P 2 1 2 1 2 2 2 2 2 P 2 P d. The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset, so, for a portfolio of Asset 1 and Asset 3: E(RP) = w1E(R1) + w3E(R3) E(RP) = .50(.1750) + .50(.1750) E(RP) = .1750 or 17.50% The variance of a portfolio of two assets can be expressed as:  = w  + w  + 2w1w3131,3  = .502(.04032) + .502(.04032) + 2(.50)(.50)(.0403)(.0403)(–1)  = .000000 Since the variance is zero, the standard deviation is also zero. e. The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset, so, for a portfolio of Asset 2 and Asset 3: E(RP) = w2E(R2) + w3E(R3) E(RP) = .50(.1750) + .50(.1750) E(RP) = .1750 or 17.50% The variance of a portfolio of two assets can be expressed as:  = w  + w  + 2w2w3232,3  = .502(.04032) + .502(.04032) + 2(.50)(.50)(.0403)(.0403)(–.3846)  = .000500 And the standard deviation of the portfolio is: P = (.000500)1/2 P = .0224 or 2.24% f. As long as the correlation between the returns on two securities is below 1, there is a benefit to diversification. A portfolio with negatively correlated stocks can achieve greater risk reduction than a portfolio with positively correlated stocks, holding the expected return on each stock constant. Applying proper weights on perfectly negatively correlated stocks can reduce portfolio variance to 0. 34. a. The expected return of an asset is the sum of the probability of each return occurring times the rate of return. So, the expected return of each stock is: E(RA) = .25(–.10) + .50(.10) + .25(.20) = .0750 or 7.50% E(RB) = .25(–.30) + .50(.05) + .25(.40) = .0500 or 5.00% b. We can use the expected returns we calculated to find the slope of the Security Market Line. We know that the beta of equity A is .25 greater than the beta of Equity B. Therefore, as beta increases by .25, the expected return on a security increases by .025 (= .075 – .05). The slope of the security market line (SML) equals: 2 P 2 1 2 1 2 3 2 3 2 P 2 P 2 P 2 2 2 2 2 3 2 3 2 P 2 P SlopeSML = Rise / Run SlopeSML = Increase in expected return / Increase in beta SlopeSML = (.075 – .05) / .25 SlopeSML = .10 or 10% Since the market’s beta is 1 and the risk-free rate has a beta of zero, the slope of the Security Market Line equals the expected market risk premium. So, the expected market risk premium must be 10 percent. 35. a. A typical, risk-averse investor seeks high returns and low risks. For a risk-averse investor holding a well-diversified portfolio, beta is the appropriate measure of the risk of an individual security. To assess the two stocks, we need to find the expected return and beta of each of the two securities. Security A: Since Security A pays no dividends, the return on Security A is simply: (P1 – P0) / P0. So, the return for each state of the economy is: RRecession = (€40 – 50) / €50 = –.20 or 20% RNormal = (€55 – 50) / €50 = .10 or 10% RExpanding = (€60 – 50) / €50 = .20 or 20% The expected return of an asset is the sum of the probability of each return occurring times the rate of return. So, the expected return of the security is: E(RA) = .10(–.20) + .80(.10) + .10(.20) = .0800 or 8.00% And the variance of the equity is:  = .10(–0.20 – 0.08)2 + .80(.10 – .08)2 + .10(.20 – .08)2  = 0.0096 Which means the standard deviation is: 2 A 2 A 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Expected Return Beta Security Market Line A = (0.0096)1/2 A = .098 or 9.8% Now we can calculate the security’s beta, which is: A = (A,M)(A) / M A = (.80)(.098) / .10 A = .784 For equity B, we can directly calculate the beta from the information provided. So, the beta for equity B is: Stock B: B = (B,M)(B) / M B = (.20)(.12) / .10 B = .240 The expected return on equity B is higher than the expected return on equity A. The risk of equity B, as measured by its beta, is lower than the risk of equity A. Thus, a typical risk- averse investor holding a well-diversified portfolio will prefer equity B. Note, this situation implies that at least one of the equities is mispriced since the higher risk (beta) equity has a lower return than the lower risk (beta) equity. b. The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset, so: E(RP) = wAE(RA) + wBE(RB) E(RP) = .70(.08) + .30(.09) E(RP) = .083 or 8.30% To find the standard deviation of the portfolio, we first need to calculate the variance. The variance of the portfolio is:  = w  + w  + 2wAwBABA,B  = (.70)2(.098)2 + (.30)2(.12)2 + 2(.70)(.30)(.098)(.12)(.60)  = .00896 And the standard deviation of the portfolio is: P = (0.00896)1/2 P = .0947 or 9.47% c. The beta of a portfolio is the weighted average of the betas of its individual securities. So the beta of the portfolio is: 2 P 2 A 2 A 2 B 2 B 2 P 2 P P = .70(.784) + .30(0.24) P = .621 36. If the weights of Oil & Gas and Minerals amount to (.4 + .35 = ) .75 of the total assets, this means that the weight of Power & Industrial is .25. The weights together with the beta of each division allows us to calculate the beta of the firm. Weir = .4(1.2) + .35(.8) + .25(.6) = .91 The weight of debt in the capital structure of Weir Group is .3 and its beta is zero. This means that the beta of the equity is: Assets = .91 = .3(0) + .7(Equity) Equity = .91/.7 = 1.3 37. a. The variance of a portfolio of two assets equals:  = w  + w  + 2wAwBBCov(A,B) Since the weights of the assets must sum to one, we can write the variance of the portfolio as:  = w  + (1 – wA) + 2wA(1 – wA) Cov(A,B) To find the minimum for any function, we find the derivative and set the derivative equal to zero. Finding the derivative of the variance function, setting the derivative equal to zero, and solving for the weight of Asset A, we find: wA = [ – Cov(A,B)] / [ +  – 2Cov(A,B)] Using this expression, we find the weight of Asset A must be: wA = (.202 – .001) / [.102 + .202 – 2(.001)] wA = .8125 This implies the weight of Asset B is: wB = 1 – wA wB = 1 – .8125 wB = .1875 b. Using the weights calculated in part a, determine the expected return of the portfolio, we find: E(RP) = wAE(RA) + wBE(RB) E(RP) = .8125(.05) + .1875(0.10) E(RP) = 0.0594 2 P 2 A 2 A 2 B 2 B 2 P 2 A 2 A 2 B 2 B 2 A 2 B c. Using the derivative from part a, with the new covariance, the weight of each stock in the minimum variance portfolio is: wA = [ + Cov(A,B)] / [ +  – 2Cov(A,B)] wA = (.202 + –.02) / [.102 + .202 – 2(–.02)] wA = .6667 This implies the weight of Asset B is: wB = 1 – wA wB = 1 – .6667 wB = .3333 d. The variance of the portfolio with the weights on part c is:  = w  + w  + 2wAwBCov(A,B)  = (.6667)2(.10)2 + (.3333)2(.20)2 + 2(.6667)(.3333)(–.02)  = .0000 Because the equities have a perfect negative correlation (–1), we can find a portfolio of the two equities with a zero variance. 38. The formula for beta is if  = .8, correlation = .7, Joos = .30 then substituting the values into the formula gives: M = .2625 = 26.25% 39. The key to completing this solution is to create a worksheet with dynamic cell references. First write out the correlation matrix with starting weights for each asset. These weights are arbitrary for now, but should add to 1. Crew Gold GGS Marine Harvest Variance Standard Deviation 0.4 0.5 0.1 Crew Gold 0.4 1 0.4 0.45 168 12.96148 GGS 0.5 0.4 1 -0.09 231 15.19868 Marine Harvest 0.1 0.45 -0.09 1 433 20.80865 2 B 2 A 2 B 2 P 2 A 2 A 2 B 2 B 2 P 2 P 2 Cov( , ) ( )  =  i M i M R R R bi = Cov(Ri, RM ) s2(RM ) = rsJoos sM =.8 = .7(.3) sM risk free 6 Now calculate the different components of the portfolio variance using these weights: Crew Gold GGS Marine Harvest Crew Gold 26.88 15.7598 4.854797 GGS 15.7598 57.75 -1.42319 Marine Harvest 4.854797 -1.42319 4.33 Variance 127.3428 The formulae used for these calculations are presented below: We now use solver to minimise the variance of the portfolio. The solution, with respective weights in bold, is given below: 40. First, we must calculate the returns on Admiral Group and the FTSE 100 Index Date Admiral Group FTSE 100 Mar-12 10.21% -1.75% Feb-12 14.45% 3.34% Jan-12 10.45% 1.96% Dec-11 -7.64% 1.22% Nov-11 -21.76% -0.70% Oct-11 -6.65% 8.11% Sep-11 -7.47% -4.93% Aug-11 -11.88% -7.23% Jul-11 -6.74% -2.19% Jun-11 -3.60% -0.74% May-11 1.83% -1.32% Apr-11 8.88% 2.73% The easiest way to calculate the beta using data is through Excel’s =SLOPE function. The beta for the full period is 0.91896. The beta for the period, March 2011 to September 2011, is 1.9806. The beta for the period, October 2011 to March 2012, is 0.07872. According to the analysis, the risk of Admiral Group has gone down between the first half of the sample period and the second half. It is not evidence against CAPM, only evidence that something may have changed in the company’s operations over this time. Chapter 10 case study A job at west coast yachts, Part 2 1. There should be little, if any, money allocated to the company equity. The principle of diversification indicates that an individual should hold a diversified portfolio. Investing heavily in company equity does not create a diversified portfolio. This is especially true since income also comes from the company. If times get bad for the company, employees face layoffs, or reduced work hours. So, not only does the investment perform poorly, but income may be reduced as well. We only have to look at employees of Enron or WorldCom to see the potential for problems with investing in company equity. At most, 5 to 10 percent of the portfolio should be allocated to company equity. 2. This is not the portfolio with the least risk. By adding equities, a riskier asset, the overall risk of the portfolio will decline. This will be demonstrated in the next questions. 3. We can use the equations for the expected return of the portfolio, and the portfolio standard deviation, that is: E(RP) = wEE(RE) + wDE(RD) P = (w  + w  + 2wEwDEDD,E)1/2 Using these equations and equity portfolio weights from zero to 100 percent at intervals of 10 percent, we get the following portfolio expected returns and standard deviations: Weight of equity fund Portfolio E(R) Portfolio standard deviation 0% 9.67% 10.8300% 10% 9.89% 10.2708% 20% 10.11% 9.9490% 30% 10.32% 9.8878% 40% 10.54% 9.0920% 50% 10.76% 10.5461% 60% 10.98% 11.2198% 70% 11.20% 12.0765% 80% 11.41% 13.0802% 90% 11.63% 14.1998% 100% 11.85% 15.4100% 2 E 2 E 2 D 2 D The graph of the opportunity set of feasible portfolios will look like the following: 4. Now we can use Solver to maximize this expression by changing the weight of equity input cell. The constraint is that the standard deviation of the portfolio is equal to the standard deviation of the bond fund. Using Solver, the weight of the large cap stock fund and bond fund in this portfolio is: wE = .5459 wD = .4541 So, the expected return and standard deviation of this portfolio is: E(R) = .5459(.1185) + .4541(.0967) E(R) = .1086 or 10.86%  = [.54592(.1541)2 + .45412(.1083)2 + 2(.5459)(.4541)(.1541)(.1083)(.27)]1/2  = .1083 or 10.83% This is the exact same standard deviation as the bond fund, but the expected return is over one percent higher. 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 0.0000% 5.0000% 10.0000% 15.0000% 20.0000% Portfolio Expected Return Portfolio Standard Deviation 5. To find the weights of each asset in the minimum variance portfolio, we begin with the equation for the variance of the portfolio. Using S to represent the large company fund and B to represent the bond fund, the variance of a portfolio of two assets equals:  = w  + w  + 2wSwBSBS,B Since the weights of the assets must sum to one, we can write the variance of the portfolio as:  = w  + (1 – wS)2 + 2wS(1 – wS)SBS,B To find the minimum for any function, we find the derivative and set the derivative equal to zero. Finding the derivative of the variance function, setting the derivative equal to zero, and solving for the weight of the stock fund, we find: wS = ( – S,B) / ( +  – 2 S,B) Using this expression, we find the weight of the equity fund, must be: wS = (.10832 –0.1541 X 0.1083 X.27) / [.15412 + .10832 – 2 X 0.1541 X 0.1083 (.27)] wS = .2729 This implies the weight of the bond fund is: wB = 1 – wS wB = 1 – .2729 wB = .7271 The expected return of this portfolio is: E(R) = .2729(.1185) + .7271(.0967) E(R) = .1026 or 10.26% 2 P 2 S 2 S 2 B 2 B 2 P 2 S 2 S 2 B 2 B SB 2 S 2 B SB The variance of the portfolio is:  = w  + w  + 2wSwBSBS,B  = (.27292)(.15412) + (.72712)(.10832) + 2(.2729)(.7271)(.1541)(.1083)(.27)  = .009758 And the standard deviation is:  = .0097581/2  = .09878 or 9.88% With these returns and variances, the minimum variance portfolio is important because no investor would ever hold a portfolio with a greater weight in bonds. If an investor increases the weight of bonds in the portfolio, the risk of the portfolio increases and the expected return decreases. The result is illustrated in Question 4. 6. We can find the Sharpe optimal portfolio by using Solver. To use Solver, we input the Sharpe ratio in a cell. The Sharpe ratio is: Sharpe ratio = We also need to recognize that the weight of debt in the portfolio is one minus the weight of equity. Substituting the equations for the expected return of the portfolio and the standard deviation of the portfolio, we get: Sharpe ratio = Now we can use Solver to maximize this expression by changing the weight of equity input cell. The question requires an estimate of the risk-free rate and students will be expected to do their 2 P 2 S 2 S 2 B 2 B 2 P 2 P σ E(R) − R f 1 2 E E E D E,D 2 D 2 E 2 E 2 E E E E D f (w σ (1 w ) σ 2w (1 w )σ σ ρ ) w E(R ) (1 w )E(R ) R + − + − / + − − own research in finding this. In our example, we will assume that the risk-free rate is 3.8%. Doing so, we find the weight of equity in the Sharpe optimal portfolio is 39.73 percent. This question can also be solved directly. The goal is to maximize the Sharpe ratio, so we can use the expression for the Sharpe ratio, set the derivative equal to zero, and solve for the weight of equity (or debt). Doing so, the resulting expression for the weight of equity in the Sharpe optimal portfolio is: wE = Using this equation, we find the weight of equity in the Sharpe optimal portfolio is: wE = wE = .3973 and the weight of debt is: wD = 1 – .3973 wD = .6027 So, the expected return and standard deviation of the Sharpe optimal portfolio is: E(R) = .3973(.1185) + .6027(.0967) E(R) = .1054 or 10.54%  = [.39732(.1541)2 + .60272(.1083)2 + 2(.3973)(.6027)(.1541)(.1083)(.27)]1/2  = .1008 or 10.08% E f D f E D E,D 2 D f E 2 E f D D f E D E,D 2 E f D [E(R ) R ]σ [E(R ) R ]σ [E(R) R E(R) R ]σ σ ρ [E(R ) R ]σ [E(R ) R ]σ σ ρ − + − − − + − − − − [.1185 .0380].1083 [.0967 .0380].1541 [.1185 .0380 .0967 .0380](.1541)(.1083)(.27) [.1185 .038].1083 [.0967 .0380](.1541)(.1083)(.27) 2 2 2 − + − − − + − − − − The Sharpe ratio of the Sharpe optimal portfolio is: Sharpe ratio = Sharpe ratio = .6681 The Sharpe optimal portfolio is the best risky portfolio for all investors because it delivers a greater reward-to-risk ratio than any other portfolio. If a line is drawn from the risk-free rate to the Sharpe optimal portfolio, it shows the best combination of portfolios available to any investor. Investors can change the level of risk by altering the percentage of their investment in the risk-free asset and the Sharpe optimal portfolio. This line is the Security Market Line. .1008 .1054 − .0380 Solution Manual for Corporate Finance David Hillier, Stephen Ross, Randolph Westerfield, Jeffrey Jaffe, Bradford Jordan 9780077139148